奇异半线性椭圆方程的非径向正整体解

POSITIVE NONRADIAL ENTIRE SOLUTIONS OF SINGULAR SEMILINEAR ELLIPTIC EQUATIONS

研究如下N维奇异半线性椭圆方程△u+f(x,u)=0,x∈R^N(N≥3),其中函数f:R^N×R_+→R_+连续,在u=0有奇异性;采用上-下解方法给出该方程具有满足如下性质的有界正整体解u的条件:u∈C_(loc)^(2+θ)(R^N)使得■u(x)=0且u(x)≥εmin{1,|x|^(2-N)},其中ε>0是常数;并证明:若条件添加"f关于u单调不增"的限制,则这种解是唯一的.

This paper is dealt with N-dimensional semilinear elliptic equation of the form△u＋f（x,u）=0,x∈R^N（N≥3）,where f：R^N×R＋→R＋is a continuous function with a singularity at u = 0. By means of super-subsolution method, some sufficient conditions are obtained for the equation to have a positive entire solution u∈Cloc^2＋θ（R^N）such that lim｜x｜→∞u（x）=0,u（x）≥εmin{1,｜x｜^2-N}where ε 〉 0 is a constant.It is also proved that if f is non-increasing in u, then the above solution is unique.